Harmonic functions and quadratic harmonic morphisms on Walker spaces

We deal with the harmonicity of quadratic maps defined on R 4 (endowed with a Walker metric q ) to the n-dimensional semi-Euclidean space of index r , and then between local charts of two 4-dimensional Walker manifolds. We obtain here the necessary and sufficient conditions under which these maps are harmonic, horizontally weakly conformal, or harmonic morphisms with respect to q.